21st LL-Seminar on Graph Theory. Universität Klagenfurt und Montanuniversität Leoben April 26 28, 2004

Transcription

1 21st LL-Seminar on Graph Theory Universität Klagenfurt und Montanuniversität Leoben April 26 28, 2004

2 Foreword More than twenty years ago graph theorists from Leoben and Ljubljana began to meet informally but regularly in the Ljubljana-Leoben Seminar on Combinatorics. The iron curtain was not so tight any more and the short distance between Ljubljana and Leoben made it easy to cooperate in those dark pre-internet times. Within the years a fruitful collaboration arose and the Seminar kept growing. Since 1989 we even have an official program for these meetings. Although organized mainly from Leoben and Ljubljana the seminar took place in many locations in Slovenia and Austria and we are grateful that the University of Klagenfurt, halfway between Leoben and Ljubljana, hosts it this time. We thank all participants for their interest and their contributions to the scientific program, and are looking forward to an interesting, productive and enjoyable meeting! Franz Rendl, Klagenfurt Wilfried Imrich, Leoben Bojan Mohar and Tomaž Pisanski, Ljubljana Sandi Klavžar and Boštjan Brešar, Maribor April

3 Acknowledgements We gratefully acknowledge financial support by the following institutions Avstrijski Znanstveni Institut Ljubljana Marie Curie Trainingsnetwork: Algorithmic Discrete Optimization (ADONET Project ) Montanuniversität Leoben Österreichische Mathematische Gesellschaft Universität Klagenfurt 2

4 Program Monday, April 26, :15 11:30 Opening and welcome addresses 11:30 12:00 Norbert Seifter Transitive Digraphs 12:00 14:00 Lunch Break 14:00 14:30 Sandi Klavžar Θ-graceful labelings of partial cubes 14:30 15:00 Josef Leydold Faber-Krahn Type Inequalities for (Non-regular) Trees 15:00 15:30 Coffee Break 15:30 16:30 Peter Mihók Additive and hereditary properties of systems of objects 16:30 18:00 Workshops 19:00 Conference Dinner Restaurant Weidenhof am See 3

5 Tuesday, April 27, :00 09:30 Igor Dukanović A Semidefinite Programming Based Heuristic for Graph Coloring 09:30 10:00 Janez Povh How to approximate the bandwidth of a graph using semidefinite programming? 10:00 10:30 Coffee Break 10:30 11:00 Sergio Cabello Planar embeddability of the vertices of a graph using a fixed point set is NP-hard 11:00 11:30 Iztok Peterin On almost-median graphs 11:30 12:00 Wilfried Imrich On edge-preserving maps of graphs 12:00 14:00 Lunch break 14:00 15:00 Problem session 15:30 Excursion - Herzogstuhl - Burg Hochosterwitz - Magdalensberg Wednesday, April 28, :00 09:30 Boštjan Brešar On Integer Domination in Graphs and Vizing-like Problems 09:30 10:00 Janez Žerovnik Weak Reconstruction of Strong Product Graphs 10:00 10:30 Coffee Break 10:30 11:00 Drago Bokal Circular chromatic number of oriented hexagonal systems 11:00 11:30 Aleksander Vesel Characterisation of the Resonance Graphs of Catacondensed Hexagonal Graphs 11:30 12:00 Petra Žigert Fibonacci cubes are the resonance graphs of fibonaccenes 12:00 End of lectures and lunch 4

6 Abstracts Abstracts are listed alphabetically with respect to Presenting Author. Circular chromatic number of oriented hexagonal systems Drago Bokal, Gašper Fijavž, Martin Juvan, Bojan Mohar and Andrej Vodopivec University of Ljubljana, FMF, Jadranska 19, 1000 Ljubljana, Slovenia The circular chromatic number of arbitrary orientation of any linear hexagonal system is determined. It depends on the existence of certain oriented substructure and is always of the form q = 5k+1 4k+1 where 1 k n 2 is an integer and n is the number of hexagons. On Integer Domination in Graphs and Vizing-like Problems Boštjan Brešar, Michael A. Henning, and Sandi Klavžar University of Maribor,FEECS, Smetanova 17, 2000 Maribor, Slovenia In this talk we consider {k}-dominating functions in graphs (or integer domination as we shall also say) that was first introduced by Domke, Hedetniemi, Laskar, and Fricke. For k 1 an integer, a function f : V (G) {0, 1,..., k} defined on the vertices of a graph G is called a {k}- dominating function if the sum of its function values over any closed neighborhood is at least k. The weight of a {k}-dominating function is the sum of its function values over all vertices. The {k}- domination number of G is the minimum weight of a {k}-dominating function of G. We studied the {k}-domination number on the Cartesian product of graphs, mostly on problems related to the famous Vizing s conjecture. Three generalizations of the inequality by Clark and Suen will be presented. Planar embeddability of the vertices of a graph using a fixed point set is NP-hard Sergio Cabello University of Ljubljana, Department of Mathematics, Jadranska 19, 1000 Ljubljana, Slovenia Let G be a graph with n vertices and let P be a set of n points in the plane. We show that deciding whether there is a planar straight-line embedding of G such that its vertices are embedded onto the points P is NP-complete, even when G is 2-connected and 2-outerplanar. This settles an open problem posed by Bose, by Brandenberg et al., and by Kaufmann and Wiese. 5

7 A Semidefinite Programming Based Heuristic for Graph Coloring Igor Dukanović and Franz Rendl University of Maribor, EPF, Razlagova 14, 2000 Maribor, Slovenia Universität Klagenfurt, Institut für Mathematik, A-9020 Klagenfurt, Austria A k-coloring of a simple undirected graph G(V, E) is a mapping x : V {1,..., k} such that (ij) E c(i) c(j) (1) This traditional graph (vertex) coloring representation induces permutation redundancy as c π, where π is any permutation of numbers 1...k, is also a k-coloring symmetric to c. We see that the graph coloring problem is in fact a partitioning problem. Therefore we introduce a coloring relation R c defined by ir c j c(i) = c(j) Any relation R is represented by a 0-1 matrix X defined by x ij = 1 irj (2) This matrix is positive semidefinite if and only if it represents an equivalence relation, and coloring relation is such. Denote by J the matrix of all ones, and let X represent any symmetric, homogeneous relation on the vertex set V satisfying (1). Then with a straightforward corollary lx J l k & matrix X represents a k-coloring χ(g) = min{l : lx J, X = X T, x ii = 1 i V, x ij = 0 (ij) E, xij {0, 1}} By dropping the integrality condition x ij {0, 1} in this NP-hard problem we obtain Lovász theta number θ(ḡ) = min{l : lx J, X = X T, x ii = 1 i V, x ij = 0 (ij) E} (3) a well known polynomial lower bound on the chromatic number of a graph. Let X be a matrix. at which an optimum of (3) is attained. Then (2) suggests that a large element x ij = 1 should be interpreted as color vertices i and j with the same color. Since θ(ḡ) and X can be computed to any fixed precision in polynomial time by semidefinite programming, this idea motivates a Karger-Motwani-Sudan like recursive heuristic for graph coloring. References [1] P. GALLINIER and J.K. HAO. Hybrid evolutionary algorithms for graph coloring. Journal of Combinatorial Optimization, 3: , [2] D. KARGER, R. MOTWANI, and M. SUDAN. Approximate graph coloring by semidefinite programming. Journal of the Association for Computing Machinery, 45: , [3] L. LOVÁSZ. On the Shannon capacity of a graph. IEEE Trans. Inform. Theory, 25:1 7,

8 Θ-graceful labelings of partial cubes Boštjan Brešar and Sandi Klavžar University of Maribor, Pf, Koroška cesta 160, Maribor, Slovenia The Ringel-Kotzig conjecture asserting that all trees are graceful remains one of the central problems in the area of graph labelings. We introduce Θ-graceful labelings of partial cubes as a natural extension of graceful labelings of trees. Several classes of partial cubes are Θ-graceful, for instance even cycles, Fibonacci cubes, and (newly introduced) lexicographic subcubes. The Cartesian product of Θ-graceful partial cubes is again such and we wonder whether in fact any partial cube is Θ-graceful. A connection between Θ-graceful labelings and representations of integers in certain number systems is also established. On edge-preserving maps of graphs Wilfried Imrich Chair of Applied Mathematics, Montanuniversität Leoben, 8700 Leoben, Austria A contraction of a graph G is a mapping f : V (G) V (G) that preserves or contracts edges, that is, whenever x, y E(G), then f(x) = f(y or f(x)f(y) E(G). Contractions are also known as weak endomorphisms or edge-preserving maps and have a plentyful variety of fixed subgraph properties. Let us just mention the well known fact that not only every automorphism, but also every contraction of a finite tree fixes a vertex or stabilizes an edge. These results have been generalized to infinite graphs by Bandelt, Chastand, Halin, Polat, Quilliot, Sabidussi, and Tits mainly with the aim to find conditions that ensure the existence of a finite fixed subgraph rather than fixed points at infinity. It is the aim of this talk to convey the spirit of the basic results and to add one about infinite median graphs. Faber-Krahn Type Inequalities for (Non-regular) Trees Josef Leydold Department for Applied Statistics and Data Processing, University of Economics and Business Administration, Augasse 2-6, 1090 Vienna, Austria The Faber-Krahn theorem states that among all bounded domains with the same volume in R n (with the standard Euclidean metric), a ball that has lowest first Dirichlet eigenvalue. Recently it has been shown that a similar result holds for (semi-)regular trees. In this article we show that such a theorem also hold for other classes of (not necessarily non-regular) trees. However, for these new results no couterparts in the world of the Laplace-Beltrami-operator on manifolds are known. 7

9 Additive and hereditary properties of systems of objects Peter Mihók Department of Applied Mathematics, Faculty of Economics, Technical University, B. Němcovej, Košice, Slovak Republic, and Mathematical Institute, Slovak Academy of Science, Grešákova 6, Košice, Slovak Republic We use the basic elementary notions of category theory. A concrete category C is a collection of objects and arrows called morphisms. An object in a concrete category C is a set with structure. We will denote the ground-set of the object A by V (A). The morphism between two objects is a structure preserving mapping. Obviously, the morphisms of C have to satisfy the axioms of the category theory. The natural examples of concrete categories are: Set of sets, FinSet of finite sets, Graph of graphs, Grp of groups, Poset of partially ordered sets with structure preserving mappings, called homomorphisms of corresponding structures. For example, a simple finite hypergraph H = (V, E) can be considered as a system of its hyperedges E = {e 1, e 2,..., e m }, where edges are finite sets and the set of its vertices V (H) is a superset of the union of hyperedges, i.e. V m i=1 e i. The following definition gives a natural generalization of graphs and hypergraphs. Let C be a concrete category. A simple system of objects of C is an ordered pair S = (V, E), where E = {A 1, A 2,..., A m } is a finite set of the objects of C, such that the ground-set V (A i ) of each object A i E is a finite set with at least two elements (i.e. there are no loops) and V m i=1 V (A i). The class of all simple systems of objects of C will be denoted by I(C). The symbols K 0 and K 1 denotes the null system K 0 = (, ) and system consisting of one isolated element, respectively. We will assume that by renaming (relabeling) the elements of the object A only, we obtain always an object A isomorphic to A in every concrete category Cl For example, graphs can be viewed as systems of objects of a concrete category of two-element sets with bijections as arrows, digraphs are systems of objects of the category of two-element posets, hypergraps are finite set systems i.e. I(F inset), etc. There are nice applications of systems of objects in information systems and computer science. A WAN network is a system on LANs, Internet is a system of WANs and the isomorphism in the category of LANs can be defined in a different way depending on the user requirements. The elements of the LANs are obviousely computers. Let us remark, that the L-structures generalizing graphs, digraphs and k-uniform hypergraphs are special systems of objects on category of relational structures. To generalize the results on generalized colourings of graphs to arbitrary simple systems of objects we need to define isomorphism of systems. We can do this in a natural way: Let S 1 = (V 1, E 1 ) and S 2 = (V 2, E 2 ) be two simple systems of objects of a given concrete category C. The systems S 1 and S 2 are said to be isomorphic if there is a pair of bijection: φ : V 1 V 2 ; ψ : E 1 E 2, such that if ψ(a 1i ) = A 2j then φ/v (A 1i ) : V (A 1i ) V (A 2j ) is an isomorphism of the objects A 1i E 1 and A 2j E 2 in the category C. The homomorphism of the systems can be defined in a similar way. The disjoint union of the systems S 1 and S 2 is the system S 1 S 2 = (V 1 V 2, E 1 E 2 ), where we assume that V 1 V 2 =. A system is said to be connected if it cannot be expressed as a disjoint union of two systems. The subsystem of S 1 induced by the set U V (S 1 ) is S 1 [U], with objects E(S 1 [U]) := {A 1i E(S 1 ) V (A 1i ) U}. S 2 is an induced-subsystem of S 1 if it is isomorphic to S 1 [U] for some U V (S 1 ). Using these definitions we can define, analogously as for graphs, that an additive induced-hereditary property of simple systems of objects of a category C is any class of systems closed under isomor- 8

10 phism, induced-subsystems and disjoint union of systems. Let us denote by MM a (C) the set of all additive induced-hereditary properties of simple systems of objects of a category C. In our talk we will consider the structure of additive hereditary properties of systems of objects. On Almost-Median Graphs Iztok Peterin Faculty of Electrical Engineering and Computer Science University of Maribor, 2001 Maribor, P.O.Box 238,Slovenia A short history of almost-median graphs will be represented. Following with two nice families of planar almost-median graphs and a new characterization of almost-median graphs. How to approximate the bandwidth of a graph using semidefinite programming? Janez Povh and Franz Rendl School of business and management, Novo mesto, Slovenia Institut fur Mathematik, University of Klagenfurt, Universitätsstraße 65-67, A-9020 Klagenfurt, Austria The bandwidth problem is an old problem from combinatorial optimization, where, given a graph G = (V, E), one looks for such a labelling Φ : V {1, 2,..., V }, which minimizes a maximal distance σ (G, Φ) = max (uv) E Φ(u) Φ(v) over all possible labellings. Let denote σ (G) = min Φ σ (G, Φ). The problem how to find the σ (G) or the optimal labelling is proven to be NP-hard problem. In the year 2000 Blum et al. proposed in [1] the semidefinite relaxation of this problem and suggested the rounding scheme, which gives the labeling Φ with σ (G, Φ ) O( n/b log n)σ (G), where b is the optimal value of the semidefinite relaxation. They argued its polynomial time complexity with ellipsoidal method, so the result seemed to have only a theoretical value. We re going to present heuristics for solving this SDP relaxation, which relies on the interior point methods and on the bundle method and gives good results in practice (much better than the theoretical guaranty does), although I can prove its polynomial time complexity only for some small classes of graphs. We re also going to present how to transform the proposed randomized algorithm into deterministic one on some specific classes of graphs (e. g. caterpillars) and how good results do we get this way. Bibliography: [1] Blum, A., Konjevod, G. Ravi, R., Vempala, S..: Semidefinite relaxations for minimum bandwidth and other vertex-ordering problems, Theor. Comput. Sci. 235 (2000), Transitive Digraphs Norbert Seifter Chair of Applied Mathematics, Montanuniversität Leoben, A-8700 Leoben, Austria For decades undirected transitive graphs have been a topic of deep investigations. Although many of the methods developed for undirected graphs can also be applied to undirected graphs (digraphs), the problems arising in the context of digraphs are quite different. In this talk we present some of these problems. 9

11 Characterisation of the Resonance Graphs of Catacondensed Hexagonal Graphs Aleksander Vesel Department of Mathematics,PeF, University of Maribor Koroška 160, SI 2000 Maribor, Slovenia The vertex set of the resonance graph of a hexagonal graph G consists of 1-factors of G, two 1-factors being adjacent whenever their symmetric difference forms the edge set of a hexagon of G. A characterization of the resonance graphs of catacondensed hexagonal graph is presented. The characterisation is the basis for the algorithm that recognizes the resonance graph of a catacondensed hexagonal graph. Moreover, the modified algorithm can be applied for recognizing the Fibonnacci cubes. Weak Reconstruction of Strong Product Graphs Blaž Zmazek and Janez Žerovnik University of Maribor, FME, Smetanova 17, 2000 Maribor, Slovenia IMFM, Jadranska 19, SI-1111 Ljubljana, Slovenia. We prove that the class of nontrivial connected strong product is weakly reconstructible. We also show that any nontrivial connected thin strong product graph can be uniquely reconstructed from each of its one-vertex-deleted deleted subgraphs. Fibonacci cubes are the resonance graphs of fibonaccenes Petra Žigert and Sandi Klavžar University of Maribor, PEF, Koroška cesta 160, 2000 Maribor, Slovenia Fibonacci cubes were introduced in 1993 and intensively studied afterwards. Fibonacci cubes are precisely the resonance graphs of fibonaccenes. Fibonaccenes are graphs that appear in chemical graph theory and resonance graphs reflect the structure of their perfect matchings. Some consequences of the main result will also be presented. 10

12 Participants C. Brand, Chair of Applied Mathematics, Montanuniversität Leoben, 8700 Leoben, Austria; B. Bresar, University of Maribor, FEECS, Smetanova 17, 2000 Maribor, Slovenia; D. Bokal, University of Ljubljana, FMF, Jadranska 19, 1000 Ljubljana, Slovenia; S. Cabello, University of Ljubljana, Department of Mathematics, Jadranska 19, 1000 Ljubljana, Slovenia; W. Dörfler, University of Klagenfurt, Universitätsstraße 65-67, 9020 Klagenfurt; I. Dukanović, University of Maribor, EPF, Razlagova 14, 2000 Maribor, Slovenia; I. Fischer, Department of Mathematics, University of Vienna, Nordbergstraße 15, 1090 Wien; G. Imrich-Schwarz, Gubattagasse 2, 8700 Leoben, Austria; W. Imrich, Chair of Applied Mathematics, Montanuniversität Leoben, 8700 Leoben, Austria; T. R. Jensen, Department of Computer Science Royal Holloway College University of London, Egham TW20 0EX, United Kingdom; S. Klavžar, University of Maribor, Pf, Koroška cesta 160, Maribor, Slovenia; J. Leydold, Department for Applied Statistics and Data Processing, University of Economics and Business Administration, Augasse 2-6, 1090 Vienna, Austria; A. Lipovec, University of Maribor, Pf, Koroška cesta 160, Maribor, Slovenia; P. Mihók, Department of Applied Mathematics, Faculty of Economics, Technical University, B. Němcovej, Košice, Slovak Republic and Mathematical Institute, Slovak Academy of Science, Grešákova 11

13 6, Košice, Slovak Republic; B. Mohar, Department of Mathematics, University of Ljubljana, Jadranska 19, 1111 Ljubljana, Slovenia; I. Peterin, Faculty of Electrical Engineering and Computer Science, University of Maribor, 2001 Maribor, P.O.Box 238, Slovenia; T. Pisanski, Department of Mathematics, University of Ljubljana, Jadranska 19, 1111 Ljubljana, Slovenia; J. Povh, School of business and management, Novo mesto, Slovenia; F. Rendl, University of Klagenfurt, Universitätsstraße 65-67, 9020 Klagenfurt; N. Seifter, Chair of Applied Mathematics, Montanuniversität Leoben, 8700 Leoben, Austria; A. Vesel, Department of Mathematics, PeF, University of Maribor, Koroška 160, 2000 Maribor, Slovenia; A. Wiegele, University of Klagenfurt, Universitätsstraße 65-67, 9020 Klagenfurt; J. Žerovnik, Faculty of Mechanical Engineering, University of Maribor, Smetanova 17, 2000 Maribor, Slovenia; P. Žigert, University of Maribor, PEF, Koroška cesta 160, 2000 Maribor, Slovenia; 12